On the rank functions of $mathcal{H}$-matroids

The notion of matroids was introduced by H. Whitney [10] in 1935 as an abstraction of the notion of linear independence in a vector space. Many researchers have studied and extended the theory of matroids (cf. [2,4,5,8,9]). In 2009, U. Faigle and S. Fujishige [1] introduced the notion of H-matroids as a general model for matroids and the greedy algorithm. They gave a characterization of H-matroids by the greedy algorithm. In this note, we give a characterization of the rank functions of H-matroids that are simplicial complexes, for any family H. Our main result is as follows.

Theorem 1.1. Let E be a finite set and let ρ : 2 E → Z ≥0 be a set function on E. Let H be a family of subsets of E with ∅, E ∈ H. Then, ρ is the rank function of an H-matroid (E, I) if and only if ρ is a normalized unitincreasing function satisfying the H-extension property.

For

Moreover, if ρ is a normalized unit-increasing set function on E satisfying the H-extension property and

) is an H-matroid with rank function ρ and I is a simplicial complex.

This note is organized as follows. Section 2 gives some definitions and preliminaries on H-matroids. In Section 3, we give a proof of Theorem 1.1 and an example which shows H-matroids that are not simplicial complexes are not characterized only by their rank functions.

Let E be a nonempty finite set and let 2 E denote the family of all subsets of E. For any family I of subsets of E, the extreme-point operator ex I : I → 2 E and the co-extreme-point operator ex A constructible family I induces a (base) rank function ρ :

The following is easily verified by definitions.

Lemma 2.1. The rank function ρ of a constructible family is normalized (i.e. ρ(∅) = 0) and satisfies the unit-increase property

Remark that, by putting X = ∅, we obtain

The restriction of I to a subset A ∈ 2 E is the family I (A) := {I ∈ I | I ⊆ A}. Note that every restriction of a constructible family is constructible.

A simplicial complex is a family I ⊆ 2 E such that X ⊆ I ∈ I implies X ∈ I. We can easily check the following lemmas on simplicial complexes. Lemma 2.2. A family I ⊆ 2 E is a simplicial complex if and only if ex I (I) = I holds for any I ∈ I.

Proof. The lemma follows from the definitions of a simplicial complex and ex I (•).

Lemma 2.3. Let I ⊆ 2 E be a simplicial complex and let X ∈ 2 E . Then, (c): Take any X ∈ 2 H and I ∈ I (H) := {I ∈ I | I ⊆ H} with X ⊆ I. Since I is a simplicial complex, X ∈ I. Since X ⊆ H, we have X ∈ I (H) .

We now recall the definitions of an H-independence system and an Hmatroid, which were introduced by Faigle and Fujishige [1]. Let E be a finite set and let H be a family of subsets of E with ∅, E ∈ H. A constructible family I ⊆ 2 E is called an H-independence system if (I) for all H ∈ H, there exists I ∈ I (H) such that |I| = ρ(H).

An H-matroid is a pair (E, I) of the set E and an H-independence system I satisfying the following property:

(M) for all H ∈ H, all the bases B of I (H) have the same cardinality |B| = ρ(H).

3 Proof of Theorem 1.1

First, we see an example which shows that H-matroids that are not simplicial complexes are not characterized by their rank functions.

Example 3.1. Let E = {1, 2, 3} and H = {∅, E}. Let

Then (E, I 1 ), (E, I 2 ), and (E, I 3 ) are H-matroids with the same rank function ρ :

Therefore, we cannot distinguish H-matroids in general by their rank functions. More generally, the following holds. In the following, we give a proof of Theorem 1.1.

Lemma 3.3. For any constructible family, there exists a simplicial complex such that their rank functions are the same.

Proof. Let I ⊆ 2 E be a constructible family. Define

Obviously each Y ∈ Max(I) is maximal in I ′ , and I ′ does not have new maximal members. Therefore Max(I) = Max(I ′ ). Note that any simplicial complex is a constructible family. By Proposition 3.2, the rank functions of I and I ′ are the same.

Lemma 3.4. Let ρ : 2 E → Z ≥0 be the rank function of an H-matroid (E, I), where I is a simplicial complex. Then ρ satisfies the H-extension property.

Proof. Take X ∈ 2 E and H ∈ H with X ⊆ H, and suppose that ρ(X) = |X| < ρ(H). By Lemma 2.3 (c), I (H) is a simplicial complex since I is a simplicial complex. Note that B(I (H) ) = Max(I (H) ) by Lemma 2.3 (a). By Lemma 2.3 (b), X ∈ I. Therefore X ∈ I (H) , and X is not a base of I (H) by (I) and (M) since ρ(X) < ρ(H). Thus there exists B ∈ I such that X B ⊆ H and |B| = ρ(H). Take any element e ∈ B \ X ⊆ H \ X. Then X ∪ {e} ∈ I since X ∪ {e} ⊆ B ∈ I and I is a simplicial complex. Hence it follows that ρ(X ∪ {e}) = |X ∪ {e}| = |X| + 1 = ρ(X) + 1.

Lemma 3.5. Let ρ : 2 E → Z ≥0 be a normalized unit-increasing function satisfying the H-extension property for some family H ⊆ 2 E with ∅, E ∈ H.

Then (E, I ρ ) is an H-matroid and I ρ is a simplicial complex.

Proof. First we show that I ρ is a simplicial complex. Take any I ∈ I ρ \ {∅} and any e ∈ I. Remark. Strict cg-matroids which were introduced by S. Fujishige, G. A. Koshevoy, and

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